Unit 3- Congruence and Proofs...Unit 3- Proofs Module 1 GEOMETRY Lesson 1: Introduction to Triangle...

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NYS COMMON CORE MATHEMATICS CURRICULUM Module 1 Unit 3- Proofs GEOMETRY

Unit 3- Congruence and Proofs

NAME ____________________________________

DATE Lesson # Page(s) Topic Homework

10/25 1 2-3 Introduction to Triangle Proofs No Homework

10/26 1 4 Intro to Proofs continued Lesson #1 HW 10/27 2 5-7 Congruence Criteria for Triangles-SAS Lesson #2 HW

10/28 3 8-9 Angles of Isosceles Triangles Lesson #3 HW 10/31 4 10-13 Congruence Criteria for Triangles-ASA/SSS Lesson #4 HW

11/1 4 Lesson 4 Continued QUIZ

11/2 5 14-17 Congruence Criteria for Triangles-SAA/HL Lesson #5 HW 11/3 6 18-21 Triangle Congruency Proofs

11/4 6 Continue Lesson 6 Lesson #6

11/7 7 22-23 Triangle Congruency Proofs II Lesson #7 11/8 8 24-26 QUIZ

Start Lesson 8 -Properties of Parallelograms

11/9 8 Finish Lesson 8 -Properties of Parallelograms

Lesson #8

11/10 9 27-28 Properties of Parallelograms II Lesson #9 11/14 10 29-32 Mid-Segment of a Triangle

11/15 10 Lesson 10 continued Lesson #10 11/16 11 33-36 Point of Concurrency Lesson #11

11/17 12 37-40 Point of Concurrency II Lesson #12

11/18 Review Ticket In 11/21 UNIT TEST

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Lesson 1: Introduction to Triangle Proofs Opening Exercise

Use your knowledge of angle and segment relationships from Unit 1, fill in the following

Property/Theorem Diagram/Key Words Statement

Definition of Right Angle

Definition of Angle Bisector

Definition of Segment Bisector

Definition of Perpendicular

Definition of Midpoint

Angles on a line

Angles at a point

Angles Sum of a Triangle

Vertical Angles

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Example 1: We are going to take this knowledge and see how we can apply it to a proof. In each of the following you are given information. You must interpret what this means by first marking the diagram and then writing it in proof form.

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Example 2:

The two most important properties about parallel lines to remember: 1.)

2.)

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Lesson 2: Congruence Criteria for Triangles—SAS

Side-Angle-Side triangle congruence criteria (SAS):

Given two triangles and so that (Side), (Angle),

(Side). Then the triangles are congruent. Consider how ABC could be mapped to A’B’C’.

Example1

In order to use SAS to prove the triangles below are congruent, draw in the missing labels:

a.) b.)

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Lesson 3: Angles of Isosceles Triangles

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What 2 Propereties to we now know about Isosceles triangles?

1.)

2.)

Once we prove triangles are congruent, we know that their corresponding parts (angles and sides) must also

be congruent. We can abbreviate this in a proof by using the reasoning of:

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

To Prove Angles or Sides Congruent:

1. Prove that the triangles are congruent

2. Use CPCTC to prove that the additional corresponding parts are congruent.

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LESSON 9: Family of Quadrilaterals

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In Unit 1 we discussed two different points of concurrency (when 3 or more lines intersect at a single point).

The two we discussed are the:

___________________ The point of concurrency of the perpendicular bisectors of a triangle.

___________________ The point of concurrency of the angle bisectors of a triangle.

Let’s find some more points of concurrency:

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LESSON 12: A closer look at points of Concurrency

Opening Exercise: Vocab Review

Match the term on the left with the definition on the right.

_____ 1) Centroid

a) The point of concurrency of the perpendicular bisectors of a triangle.

_____ 2) Mid-Segment

b) A line segment that connects the vertex of a triangle to the opposite side at a right angle.

_____ 3) Median

c) A line segment that joins the midpoints of two sides in a triangle.

_____4) Circumcenter

d) The point of concurrency in the altitudes of a triangle.

_____5) Incenter

e) The point of concurrency of the angle bisectors of a triangle.

_____ 6) Altitude

f) A line segment drawn from the vertex of a triangle to the midpoint of the opposite side.

_____7) Perpendicular Bisector

g) The point of concurrency of the medians of a triangle.

_____8) Orthocenter h) A line or segment that is perpendicular to a segment and bisects that segment.

Important Fact: A centroid splits the medians of a triangle into two smaller segments. These segments are

always in a 2:1 ratio.

Label the lengths of segments DF, GF, and EF as x, y and z respectively. Express the lengths of CF, BF and AF in

terms of x, y and z.

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Unit 3- Congruence and Proofs...Unit 3- Proofs Module 1 GEOMETRY Lesson 1: Introduction to Triangle...

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